#### Vol. 267, No. 1, 2014

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Normal states of type III factors

### Yasuyuki Kawahigashi, Yoshiko Ogata and Erling Størmer

Vol. 267 (2014), No. 1, 131–139
##### Abstract

Let $M$ be a factor of type III with separable predual and with normal states ${\phi }_{1},\dots ,{\phi }_{k},\omega$ with $\omega$ faithful. Let $A$ be a finite-dimensional ${C}^{\ast }$-subalgebra of $M$. Then it is shown that there is a unitary operator $u\in M$ such that ${\phi }_{i}\circ Adu=\omega$ on $A$ for $i=1,\dots ,k$. This follows from an embedding result of a finite-dimensional ${C}^{\ast }$-algebra with a faithful state into $M$ with finitely many given states. We also give similar embedding results of ${C}^{\ast }$-algebras and von Neumann algebras with faithful states into $M$. Another similar result for a factor of type II${}_{1}$ instead of type III holds.

 Dedicated to Masamichi Takesaki on the occasion of his eightieth birthday.
##### Keywords
von Neumann algebra, type III factor, normal state
Primary: 46L30